Well, sometimes family wants to know our source of income and make sure it is a clean source of income. Many people know that there are a lot of scams on the internet these days, and our family doesn't want us to be those people. Therefore, it is very important to let them know about the basics of what we do on the internet, so that they know too. However, as you said, it all depends on how your family perceives you. However, I feel that it is important.
Then if I told parents that I earn due to cryptocurrency price growth, they might not be very happy about it. Cryptocurrency itself does not generate money. If I have bought bitcoin @ $20k, and sold it @ $60k, that means somebody has lost $40k on trades or bad investment. I have earned because someone has lost. The fact that my success depends on someone else failure might be disappointing for parents, when they find out it is all money related.
But I would make a drama, would not make special preparation and prepare soil for a pleasant conversation with parents if I was to tell them about how I earn. If I earn honestly, then there is nothing to worry about it.